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We have an infinite sequence of random variables which are exchangeable. Could we say that each finite subsequence of this sequence is again exchangeable?

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By definition, an infinite collection of random variables is said to be exchangeable if every finite subset of those variables is exchangeable. (See definition 1.11 in Schervich's book "Theory of statistics" (p.7).

A finite collection of finite random variables is said to be exchangeable if their distribution is invariant under permutations.

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